Extended Erlang B Calculator (Caller Retry & Recall Factor)
Quantify how human redial behavior inflates offered traffic intensity, prevents classical Lost-Calls-Cleared (LCC) assumptions from holding, and leads to severe trunk under-provisioning under congestion.
The Mathematical Theory of Extended Erlang B & Subscriber Retrials
1. The Flaw in Classical Erlang B: The Myth of the Docile Caller
In 1917, Danish mathematician and telephone engineer Agner Krarup Erlang published his seminal formula for dimensioning telephone trunks, now known worldwide as Erlang B. The fundamental architectural premise of Erlang B is the Lost-Calls-Cleared (LCC) assumption: when an arriving call encounters an all-trunks-busy (ATB) state, the switch rejects the call, and the subscriber is assumed to vanish permanently from the telecommunications network.
In modern telecommunications, this assumption fails catastrophically. Human telephone callers and automated SIP dialers are not docile; when an urgent business call, customer service inquiry, or emergency connection encounters a fast-busy tone or network rejection (SIP 486 Busy Here / SIP 503 Service Unavailable), the caller immediately presses "redial." This transforms the caller from an external Poisson arrival into a closed-loop feedback generator, recycling back into the trunk group within seconds.
2. Mathematical Formulation of the Extended Erlang B Model
The Extended Erlang B (EEB) model accounts for subscriber redial psychology by introducing the Recall Factor (R), where 0 ≤ R ≤ 1 represents the probability that a blocked caller will make another call attempt within the busy hour:
- R = 0.0 (Pure Erlang B): Zero retries; all blocked calls are permanently lost (classical LCC).
- R = 0.5 (Standard Telephony): Exactly half of all blocked calls generate an immediate retry attempt.
- R → 1.0 (Persistent Redialing): Callers aggressively hit redial until they secure an open trunk channel.
The total effective offered traffic Aeff confronting the trunk group is the sum of raw first-attempt traffic Araw and the retrial stream Aretry:
Aretry = Aeff × B(m, Aeff) × R × Hr
where B(m, Aeff) is the classical Erlang B blocking probability for m channels at load Aeff, and Hr is the holding time ratio of retried attempts relative to original calls.
3. Solving the Non-Linear Contraction Mapping (Banach's Fixed-Point Theorem)
Because the blocking term B(m, Aeff) depends non-linearly on Aeff, this system cannot be solved algebraically in closed form. Instead, teletraffic engineers employ an iterative fixed-point contraction mapping:
Ak+1 = Araw + Ak × B(m, Ak) × R
Convergence Criteria: |Ak+1 - Ak| < 10−6
To ensure 100% numerical stability without 64-bit floating-point factorial overflow, B(m, Ak) is computed recursively:
B(j, A) = [ A × B(j−1, A) ] / [ j + A × B(j−1, A) ] (for j = 1, 2, ..., m)
By Banach's Fixed-Point Theorem, because the derivative of the recurrence relation is strictly bounded within [0, 1) for stable physical traffic regimes, the sequence is guaranteed to converge monotonically to a unique steady-state equilibrium.
4. The Danger of Under-Provisioning: The Congestion Cascade
Trunk groups sized strictly using classical Erlang B are chronically under-dimensioned in real-world networks. Consider an enterprise trunk sized for 20 Erlangs of raw traffic across 25 channels. Classical Erlang B predicts a modest 5.02% blocking rate.
However, with a typical 50% recall factor (R = 0.5), blocked calls immediately re-enter the trunk group, inflating offered traffic to 20.62 Erlangs. This pushes actual blocking up to 5.99%.
During unexpected traffic surges (e.g., severe weather, promotional campaigns, or disaster hotlines where R ≥ 0.85), this feedback loop triggers a teletraffic congestion avalanche: initial blocking causes massive redials, which further inflate offered traffic, causing even higher blocking. Trunk utilization approaches 100% while actual successful carried calls plummet.
5. Comparing Teletraffic Models: Erlang B vs. Extended Erlang B vs. Erlang C
- Classical Erlang B (M/M/m/m): Lost Calls Cleared (LCC). Assumes zero queuing and zero retrials. Severely underestimates blocking when human redials occur.
- Extended Erlang B: Lost Calls Retried (LCR). Assumes no centralized switch queue, but models subscriber redial feedback. Accurately reflects unqueued PSTN and SIP trunk groups.
- Erlang C (M/M/m/∞): Lost Calls Delayed (LCD). Assumes callers are held in an automated switch/ACD queue with infinite buffer space until an agent or channel becomes free.
- Engset Model: Finite subscriber sources. Used when the calling population is small relative to trunk capacity, causing traffic intensity to drop as active calls tie up callers.
Extended Erlang B vs. Classical Erlang B Reference Lookup Table
Compare blocking percentages and required channels between Classical Erlang B (R = 0) and Extended Erlang B under moderate (R = 0.5) and aggressive (R = 0.8) recall conditions at standard 1% GoS (P.01).
| Raw Traffic (Araw) | Given Trunks (m) | Classical B (R=0) | EEB (R=0.5) | EEB (R=0.8) | Trunks (1% GoS, R=0) | Trunks (1% GoS, R=0.8) |
|---|---|---|---|---|---|---|
| 5.0 Erlangs | 9 Trunks | 4.54% | 5.38% | 6.12% | 10 Trunks | 11 Trunks |
| 10.0 Erlangs | 15 Trunks | 4.70% | 5.68% | 6.57% | 18 Trunks | 19 Trunks |
| 15.0 Erlangs | 21 Trunks | 4.14% | 5.06% | 5.88% | 24 Trunks | 26 Trunks |
| 20.0 Erlangs | 27 Trunks | 3.65% | 4.49% | 5.25% | 30 Trunks | 33 Trunks |
| 30.0 Erlangs | 38 Trunks | 3.52% | 4.41% | 5.21% | 42 Trunks | 46 Trunks |
| 50.0 Erlangs | 59 Trunks | 3.38% | 4.29% | 5.11% | 64 Trunks | 70 Trunks |
| 75.0 Erlangs | 85 Trunks | 3.29% | 4.21% | 5.04% | 92 Trunks | 100 Trunks |
| 100.0 Erlangs | 111 Trunks | 3.23% | 4.17% | 5.00% | 119 Trunks | 129 Trunks |
| 150.0 Erlangs | 163 Trunks | 3.16% | 4.11% | 4.95% | 173 Trunks | 187 Trunks |